On the relation between K- and L-theory of $$C^*$$ C ∗ -algebras

Abstract

We prove the existence of a map of spectra $$\tau _A:kA \rightarrow \ell A$$ between connective topological K-theory and connective algebraic L-theory of a complex $$C^*$$ -algebra A which is natural in A and compatible with multiplicative structures. We determine its effect on homotopy groups and as a consequence obtain a natural equivalence $$KA\left[ \tfrac{1}{2}\right] \xrightarrow {\simeq } LA\left[ \tfrac{1}{2}\right] $$ of periodic K- and L-theory spectra after inverting 2. We show that this equivalence extends to K- and L-theory of real $$C^*$$ -algebras. Using this we give a comparison between the real Baum–Connes conjecture and the L-theoretic Farrell–Jones conjecture. We conclude that these conjectures are equivalent after inverting 2 if and only if a certain completion conjecture in L-theory is true.